Newton Polygons of Hecke Operators

In this computational paper we verify a truncated version of the Buzzard-Calegari conjecture on the Newton polygon of the Hecke operator \(T_2\) for all large enough weights. We first develop a formula for computing \(p\)-adic valuations of exponential sums, which we then implement to compute \(2\)-...

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Bibliographic Details
Published in:arXiv.org 2020-09
Main Authors: Chiriac, Liubomir, Jorza, Andrei
Format: Article
Language:English
Subjects:
Operators
Polygons
Polynomials
Online Access:Get full text
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Summary:In this computational paper we verify a truncated version of the Buzzard-Calegari conjecture on the Newton polygon of the Hecke operator \(T_2\) for all large enough weights. We first develop a formula for computing \(p\)-adic valuations of exponential sums, which we then implement to compute \(2\)-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard-Calegari polynomial has a vertex at \(n\leq 15\), then it agrees with the Newton polygon of \(T_2\) up to \(n\).
ISSN:2331-8422